Problem: You have found the following ages (in years) of 5 turtles. The turtles are randomly selected from the 33 turtles at your local zoo: $ 92,\enspace 57,\enspace 13,\enspace 107,\enspace 51$ Based on your sample, what is the average age of the turtles? What is the variance? You may round your answers to the nearest tenth.
Answer: Because we only have data for a small sample of the 33 turtles, we are only able to estimate the population mean and variance by finding the sample mean $({\overline{x}})$ and sample variance $({s^2})$ To find the sample mean , add up the values of all $5$ samples and divide by $5$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{5}} x_i}{{5$ To compensate for this underestimation, rather than simply averaging the squared deviations from the mean , we total them and divide by $n - 1$ $ {s^2} = \dfrac{\sum\limits_{i=1}^{{n}} (x_i - {\overline{x}})^2}{{n - 1}} $ $ {s^2} = \dfrac{{784} + {49} + {2601} + {1849} + {169}} {{5 - 1}} $ $ {s^2} = \dfrac{{5452}}{{4}} = {1363\text{ years}^2} $ We can estimate that the average turtle at the zoo is 64 years old. There is a variance of 1363 years $^2$.